How to stop a random walk

Question

plot(0:70,0:70, type="n", xlab="X", ylab="Y")

x<-40
y<-40

x2<-60
y2<-60

points(x, y, pch=16, col="red", cex=1.5)
points(x2, y2, pch=16, col="green", cex=1.5)

for (i in 1:10000){
    xi<-sample(c(1,0,-1),1)
    yi<-sample(c(1,0,-1),1)
    x2i<-sample(c(1,0,-1),1)
    y2i<-sample(c(1,0,-1),1)
    lines(c(x,x+xi),c(y,y+yi))
    lines(c(x2,x2+x2i),c(y2,y2+y2i), col="red")
    x<-x+xi
    y<-y+yi
    x2<-x2+x2i
    y2<-y2+y2i
    if(x2==x && y==y2) {
        break
    }
}

I have this random walk with two lines and I need it to stop when the two lines meet.

First, I drew an empty plot and two start points for the lines. Then I have this for loop for the lines to move, to draw them on the plot and to get the new start points for the next iteration.

I tried to make it stop when the lines meet using: if(x2==x && y==y2) { break } but the lines only stop if they are on the same point and at the same time (in the same iteration) and I need them to stop if one of them crosses the other. If one crosses any point already draw for the other line. I think the problem is that the points already drawn are not saved anywhere so I cannot compare them with the points of the lines. Maybe I need to save the points out of the loop? Someone knows how to stop it?

Basically, you should be storing all your vertices in a matrix inside your loops. in that way you can check the newest point(s) against all previous points relatively easily. However, checking that you have crossed any **line** is not satisfied by simply checking endpoints. Can you restate the exact stopping condition (same vertex, or two line segments intersecting), so we can suggest the math formulas to use? — Carl Witthoft, Jan 18 '14 at 19:20
Avoid using for loop, rather use vectors. Then you will have objects with all the coordinates, walk runs, etc. stored for let say 10000 runs (ie rows), like you use it in the for loop. It will be also faster. — Petr Matousu, Jan 19 '14 at 01:14

jlhoward · Answer 1 · 2014-01-19T11:31:55.180

N      <- 10000
D      <- 1
coef.1 <- matrix(NA,N,2)
coef.2 <- matrix(NA,N,2)
path.1 <- matrix(NA,N,2)
path.2 <- matrix(NA,N,2)
path.1[1,] <- c(40,40)
path.2[1,] <- c(60,60)
d.start    <- sqrt(sum((path.1[1,]-path.2[1,])^2))
ch <- "."
set.seed(1)
system.time({
  for (i in 2:N){
    if (i%%50==0) cat(ch)
    path.1[i,] <- path.1[i-1,] + sample(-D:D,2)
    path.2[i,] <- path.2[i-1,] + sample(-D:D,2)
    coef.1[i,] <- get.line(path.1[(i-1):i,])
    coef.2[i,] <- get.line(path.2[(i-1):i,])
    r.1 <- sqrt(max(rowSums((path.1[1:i,]-path.1[1,])^2)))
    r.2 <- sqrt(max(rowSums((path.2[1:i,]-path.2[1,])^2)))
    if (r.1+r.2 < d.start) next  # paths do not overlap
    ch <- "1"
    d.1 <- sqrt(min(rowSums((path.2[1:i,]-path.1[1,])^2)))
    d.2 <- sqrt(min(rowSums((path.1[1:i,]-path.2[1,])^2)))
    if (d.1>r.1 & d.2>r.2) next
    ch <- "2"
    cross <- sapply(2:i,
               function(k){seg.intersect(path.2[(k-1):k,],path.1[(i-1):i,],k)})
    if (any(cross)) break
    cross <- sapply(2:i,
               function(k){seg.intersect(path.1[(k-1):k,],path.2[(i-1):i,],k)})
    if (any(cross)) break
  }
})
# 11111111112222222222222222222222
#    user  system elapsed 
# 1016.82    0.13 1024.18
print(paste("End at Step: ",i))
# [1] "End at Step:  1624"
plot(0:100,0:100, type="n", xlab="X", ylab="Y")
points(path.1[1,1],path.1[1,2], pch=16, col="red", cex=1.5)
points(path.2[1,1],path.2[1,2], pch=16, col="green", cex=1.5)
lines(path.1[1:i,])
lines(path.2[1:i,],col="red")

As @CarlWitthoft points out, at each step you must check all the previous line segments for crossings. This creates a serious problem, because at each new step, i, there are 2*(i-1) tests for crossings. So, if you get to a crossing at step k, there will have been 2*k*(k+1) tests. If k ~O(10000), then there can be potentially 100MM tests.

To make this more efficient, we store not only the two new points at each step, but also the slope and intercept of the newly created line segments. This avoids recalculating slope and intercept for all prior line segments at each step. In addition, we calculate the path radius, r, for each path at each step. This is the distance between the starting point and the point on the path farthest from the starting point. If the distance between the path starting point and the closest point on the other path is larger than the path radius, there can be no crossings and we can skip the individual segment comparisons for this step.

Your problem is interesting for other reasons. The normal way to test for crossings is to determine if the intersection between the two lines is on either segment. This is cumbersome but straightforward. However there are many special cases: Are the lines parallel? If so, are they coincident? If so do the segments overlap? What about vertical lines (slope=Inf). Because you set the increment to a random integer on [-1,1], all of these possibilities are quite likely to happen eventually in a path with 10000 steps. So the function seg.intersect(...) above has to account for all these possibilities. You would think there is a function in R that does that, but I couldn't find one, so here is a (messy) version:

get.line <- function(l) {        # returns slope and intercept 
  if (diff(l)[1]==0) return(c(Inf,NA))
  m <- diff(l)[2]/diff(l)[1]
  b <- l[1,2]-m*l[1,1]
  return(c(m,b))
}
is.between <- function(x,vec) {  # test if x is between values in vec
  return(x>=range(vec)[1] & x<=range(vec)[2])
}
special.cases = function(l1,l2, coeff) {
  # points coincide: not a line segment!
  if (rowSums(diff(l1)^2)==0 | rowSums(diff(l2)^2)==0) return(c(NA,FALSE))
  # both lines vertical
  if (is.infinite(coeff[1,1]) & is.infinite(coeff[2,1])) {
    if (l1[1,1]!=l2[1,1]) return(c(NA,FALSE))
    t1 <- is.between(l1[1,2],l2[,2]) | is.between(l1[2,2],l2[,2])
    t2 <- is.between(l2[1,2],l1[,2]) | is.between(l2[2,2],l1[,2])
    return(c(NA,t1|t2))
  }
  # only l1 is vertical
  if (is.infinite(coeff[1,1]) & is.finite(coeff[2,1])) {
    x <- l1[1,1]
    y <- c(x,1) %*% coeff[2,]
    return(c(x,y))
  }
  # only l2 is vertical
  if (is.finite(coeff[1,1]) & is.infinite(coeff[2,1])) {
    x <- l2[1,1]
    y <- c(x,1) %*% coeff[1,]
    return(c(x,y))
  }
  # parallel, non-coincident lines
  if (diff(coeff[,1])==0 & diff(coeff[,2])!=0) return(c(NA,FALSE))
  # parallel, coincident lines
  if (diff(coeff[,1])==0 & diff(coeff[,2])==0) {
    x <- l1[1,1]
    y <- l1[1,2]
    return(c(x,y))
  }
  # base case: finite slopes, not parallel
  x <- -diff(coeff[,2])/diff(coeff[,1])
  y <- c(x,1) %*% coeff[1,]
  return(c(x,y))   
}
seg.intersect <- function(l1,l2,i){
  pts   <- list(l1,l2)
  coeff <- rbind(coef.1[i,],coef.2[i,])
  z <- special.cases(l1,l2, coeff)
  if (is.na(z[1])) return (z[2])
  #  print(coeff)
  #  print(z)
  found <- do.call("&",
    lapply(pts,function(x){is.between(z[1],x[,1]) & is.between(z[2],x[,2])}))
  return(found)
}

How to stop a random walk

1 Answers1